What form do I need to fill out if I have a class III weapon and I am moving within the state? I have a copy of the form I filled out before I moved here from Florida, but I don't know if it is the same form as I need to fill out if I am just changing residences within Georgia.

A registered possessor of an NFA firearm, other than an FFL/SOT, may not lawfully transport in interstate or foreign commerce any destructive device, machinegun, short-barreled shotgun, or short-barreled rifle, without prior written approval of ATF, specifically the NFA Branch. Approval for the transportation may be obtained by either a written request, or an approved application filed with ATF on Form 5320.20, Application to Transport Interstate or to Temporarily Export Certain NFA Firearms. Please note that applications to transport NFA firearms will be approved only if consistent with all State and local laws.
[18 U.S.C. § 922(a)(4); 27 C.F.R. § 478.28]

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However, they have requested that they be notified of any changes of address so the registry can be updated. That notification can be a Form 20, or simply a letter with all the pertinent details.